I use my laptop (Dell Inspiron) most of the time through the mains without the battery without any problems. I only do so because I heard that keep recharging the battery, even for shot periods can shorten its life. Whether this is true or not, I don't know.

Why you want to run your laptop like that? Any problem with the battery or charging process? Most of the laptops will work without the battery. But as memphis said if its a part of charging circuit/ power-up then you will be in trouble. whats the make n' model of your laptop?

I use 2 laptops: Gateway Solo 9500 and Fujitsu TX 765 without battery for years. My reason is the weight and the fact at my job place the AC is...continual. No problem at all. Yes the Lithium-Ion batteries has a longer life at a few recharges.